L11a435

L11a434

L11a436

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Link Presentations

[edit Notes on L11a435's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,12,19,11 X14,8,15,7 X8,14,9,13 X22,20,13,19 X20,16,21,15 X16,22,17,21 X12,18,5,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -9}, {5, -4, 7, -8, 9, -3, 6, -7, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{(v-1) \left(-u v w^3+4 u v w^2-4 u v w+2 u v+u w^3-2 u w^2+2 u w+2 v^2 w^2-2 v^2 w+v^2+2 v w^3-4 v w^2+4 v w-v\right)}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+3 q^7-7 q^6+13 q^5-18 q^4+21 q^3-20 q^2+19 q-13+9 q^{-1} -3 q^{-2} + q^{-3} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ - a^{-8} +3 z^2 a^{-6} - a^{-6} z^{-2} +2 a^{-6} -3 z^4 a^{-4} -2 z^2 a^{-4} +4 a^{-4} z^{-2} +4 a^{-4} +z^6 a^{-2} +a^2 z^2-5 z^2 a^{-2} -5 a^{-2} z^{-2} +a^2-9 a^{-2} -2 z^4-z^2+2 z^{-2} +3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +4 z^9 a^{-1} +8 z^9 a^{-3} +4 z^9 a^{-5} +16 z^8 a^{-2} +16 z^8 a^{-4} +6 z^8 a^{-6} +6 z^8+3 a z^7+4 z^7 a^{-1} +3 z^7 a^{-3} +7 z^7 a^{-5} +5 z^7 a^{-7} +a^2 z^6-46 z^6 a^{-2} -38 z^6 a^{-4} -5 z^6 a^{-6} +3 z^6 a^{-8} -15 z^6-6 a z^5-32 z^5 a^{-1} -48 z^5 a^{-3} -29 z^5 a^{-5} -6 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+52 z^4 a^{-2} +35 z^4 a^{-4} -z^4 a^{-6} -5 z^4 a^{-8} +18 z^4+3 a z^3+39 z^3 a^{-1} +68 z^3 a^{-3} +37 z^3 a^{-5} +3 z^3 a^{-7} -2 z^3 a^{-9} +3 a^2 z^2-44 z^2 a^{-2} -22 z^2 a^{-4} +4 z^2 a^{-6} +3 z^2 a^{-8} -18 z^2-23 z a^{-1} -41 z a^{-3} -21 z a^{-5} -2 z a^{-7} +z a^{-9} -a^2+23 a^{-2} +14 a^{-4} - a^{-8} +10+5 a^{-1} z^{-1} +9 a^{-3} z^{-1} +5 a^{-5} z^{-1} + a^{-7} z^{-1} -5 a^{-2} z^{-2} -4 a^{-4} z^{-2} - a^{-6} z^{-2} -2 z^{-2} }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     2   2 13                   5 1   -4 11                 8 2     6 9               11 6       -5 7             10 7         3 5           10 11           1 3         9 10             -1 1       6 12               6 -1     3 7                 -4 -3     6                   6 -5 1 3                     -2 -7 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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